The Mathematics of Music (MAT 1631)
Instructor: Dr.
Rachel Hall
Office: 229
Barbelin
Office
Hours: T 10:3011:30, W 9:3010:30, F 121
Telephone:
(610) 6603096 (Office)
Email: rhall@sju.edu
Course Description: Music has many connections to mathematics. The ancient Greeks discovered that chords with a pleasing sound are related to simple ratios of integers. The mathematics of rhythm has also been studied for centuries—in fact, ancient Indian writers discovered the Fibonacci sequence in the rhythms of Sanskrit poetry. Other connections between math and music include the equations describing the sounds of musical instruments, the mathematics behind digital recording, the use of symmetry and group theory in composition, the exploration of patterns by African and Indian drummers, the application of chaos theory to modeling the behavior of melodies, and the representation of chords by exotic geometric objects called orbifolds. Along the way, we discuss the role of creativity in mathematics and the ways in which mathematics has inspired musicians.
Course Objectives: This course introduces a number of mathematical topics and investigates their applications to the analysis of music. I intend to use the medium of musical analysis to (1) explore mathematical concepts such as Fourier series and tilings that are not covered in other math courses, and (2) introduce topics such as group theory and combinatorics covered in more detail in upperlevel math courses. The course is not proofsbased. Students will complete a semesterlong project that explores one aspect of the course in depth.
Prerequisite: Calculus II and some musical training (Music 1511 or the equivalent). Students with exceptional performance in Calculus I (or AP) and musical training will be admitted on a casebycase basis.
Texts: Daniel Levitin, This is Your Brain on Music, Dutton, 2006. David Benson, Music: a Mathematical Offering, Cambridge University Press, 2006. This text is available free online or for purchase at the bookstore. (Warning: the page numbers in the online version are different from the printed one. I will refer to the numbers in the printed book in class.)
Quizzes: There will be a 15minute quiz given in class
every Wednesday when there is not a problem set due, giving ten quizzes. Quizzes are based on homework “drill”
problems, readings, and class notes.
There are no makeup quizzes, but your lowest two grades will be dropped.
Problem sets: There will be four problem sets, due February 6, February 29, March 28, and April 18 (all Fridays). Strict academic honesty policies apply.
Project: Projects explore one or more aspects of the course in depth. The final project includes three presentations and a written paper plus supporting materials as appropriate. Anyone who submits a project that I judge to be of publishable quality and submits it to a journal or competition will receive an A for the course.
Grading scale:
Grades are calculated out of 500
points:
Four problem
sets 
50 points
each 
Final project 
200 points 
Ten quizzes
(lowest two dropped) 
10 points
each 
Class
preparation and participation 
20 points 
The cutoffs are
466 A, 450 A, 433 B+, 416 B, 400 B, 366 C, 350 C, 333 D+, 300 D, and below
300 F. Grades may be curved at the
end of the semester.
Academic
Honesty:
Dishonesty includes cheating on a test, falsifying data, misrepresenting the
work of others as your own (plagiarism), and helping another student cheat or
plagiarize. At the very least, an academic honesty infraction will result in
the filing of a violation report and a grade of zero on that particular
assignment; serious or repeated infractions of the Academic Honesty policy will
result in failure of the course. For complete information about the
University’s policy on Academic Honesty, consult the Student Handbook
20072008.
Attendance: Class attendance is
mandatory. Although I do not have a rigid cut policy, anyone who has
missed lots of classes and is doing poorly in the course should not expect much
sympathy from me. If you do miss a class, it is your responsibility to
obtain the notes and assignments from another. (There are no makeup quizzes.)
Students with Disabilities: Policies for students with disabilities are posted on the course web site.
Schedule (subject to change):
Music 
Math 
Poetic meter 
Intro to combinatorics;
recursion and iteration; de Bruijn sequences 
Analysis of drumming
patterns in world music 
Phylogenetic analysis of
binary sequences; asymmetric rhythms; tiling canons 
Review of music notation 

Problem set 1, due February 6 

The physics of musical
instruments, tuning 
Wave equation and intro to
PDE’s, Fourier series, beats 
Digital recording and
synthesis 
Intro to signal
processing: sampling, DFT,
aliasing 
Project proposal presentations (date TBA) Problem set 2, due February 29 

Scale theory Melodic transformations
(inversion, retrograde, etc) Harmonic transformations
(transposition, inversions) 12tone theory NeoRiemannian set theory 
Equivalence relations;
group theory; Burnside’s lemma 
Project progress report presentations (date TBA) Problem set 3, due March 28 

Harmony, counterpoint,
voice leading 
Tilings, quotient spaces
(orbifolds), elementary topology 
Problem set 4, due April 18 

(Optional) Melodic models 
1/f noise 
Project consultations 

Final project presentations Friday, May 2, 11:30 am  1:30 pm 